arXiv:cond-mat/0406627 · Physical Review Letters

Atom-Dimer Scattering for Confined Ultracold Fermion Gases

Christophe Mora, Reinhold Egger, Alexander O. Gogolin, A. Komnik

Published Oct 22, 2004• 51 citations

Abstract

We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<<-1, however, the potential has long range, with a(ad)>0 and b(ad)>>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.

DOI: 10.1103/physrevlett.93.170403

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arXiv:cond-mat/0406627 · Physical Review Letters

Atom-Dimer Scattering for Confined Ultracold Fermion Gases

Christophe Mora, Reinhold Egger, Alexander O. Gogolin, A. Komnik

Published Oct 22, 2004• 51 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.93.170403

Related papers

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a

and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find

{b}_{\mathit{ad}}=0

and

{a}_{\mathit{ad}}

is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal:

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter

{R}^{*}

related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.

We solve the two-body problem of ultracold heteronuclear atoms in a quasi-two-dimensional (quasi-2D) geometry. The quasi-2D confinement is realized by a harmonic trap along the longitudinal (

z

-) direction, with different trap frequencies for the two atoms, as in many current experiments on ultracold heteronuclear gases. As a consequence, the longitudinal center-of-mass (CoM) motion is coupled to the relative motion, which significantly complicates the two-body problem. We solve this problem exactly and derive the 2D scattering length

a_{\rm 2D}

, the 2D effective range parameter

R_{\rm 2D}

, and the bound-state energies, as functions of the

s

-wave scattering length and effective range of the two atoms in free three-dimensional (3D) space. We show that multiple 2D scattering resonances can be induced by the coupling between the longitudinal CoM and relative motion. Around these resonances,

a_{\rm 2D}

varies rapidly with the 3D scattering parameters, while

R_{\rm 2D}

is strongly enhanced. Since the effective pairwise interaction in quasi-2D ultracold gases is determined by i.e., the two-body scattering amplitudes and bound-state energies, our results can be used for manipulating the effective 2D interatomic interaction in quasi-2D ultracold heteronuclear gases by tuning the confinement frequencies and the 3D scattering parameters.

s

-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.